ok it is an old ski, so first thing is to clean the fuel system completely clean the fuel tank really good. Replace the fuel lines and as someone said bypass the fuel selector totally. Hook up a line from the reserve pickup on the tank to the fuel pump on the front of the carb, inspect the filter and clean it also. check the pulse tube on the fuel pump.
Change the return fuel line I like clear tubing so I can see the fuel traveling through the line, I also like to restrict the fuel in the return line so it is held back somewhat . Some people use a carb jet in the line but you can just use a hose clamp and tighten it enough that when you are blowing in it you feel a bit of resistance. This keeps more fuel available to the needle and seats inside the diaphragm .
You do need to know how the compression is and if the carbs are in good shape or do they need rebuilding especially the diaphragms and internal filters.
Next you need to know the condition of the plug wires and the spark, many of those older skies have deteriorated plug wire cores that will just drive you crazy trying to figure out what the issue is. There is no simple answer you need to methodically check the condition of each component or practice swimming and pushing the ski at the same time. REALLY.
